Skeleton-stabilized divergence-conforming B-spline discretizations for highly advective incompressible flow problems

TL;DR

This paper introduces a stabilization technique for divergence-conforming B-spline discretizations in incompressible flow simulations, ensuring pressure robustness, stability, and high accuracy, while effectively reducing numerical artifacts and enabling efficient large eddy simulations.

Contribution

The paper presents a novel stabilization method penalizing high-order derivative jumps, improving accuracy and stability in divergence-conforming B-spline discretizations for incompressible Navier--Stokes problems.

Findings

Achieves optimal convergence rates for velocity

Eliminates spurious small-scale structures

Effective as an Implicit Large Eddy Simulation method

Abstract

We consider a stabilization method for divergence-conforming B-spline discretizations of the incompressible Navier--Stokes problem wherein jumps in high-order normal derivatives of the velocity field are penalized across interior mesh facets. We prove that this method is pressure robust, consistent, and energy stable, and we show how to select the stabilization parameter appearing in the method so that excessive numerical dissipation is avoided in both the cross-wind direction and in the diffusion-dominated regime. We examine the efficacy of the method using a suite of numerical experiments, and we find the method yields optimal $\textbf{L}^2$ and $\textbf{H}^1$ convergence rates for the velocity field, eliminates spurious small-scale structures that pollute Galerkin approximations, and is effective as an Implicit Large Eddy Simulation (ILES) methodology.